A New Gap Function for Vector Variational Inequalities with an Application

Abstract

We consider a vector variational inequality in a finite-dimensional space. A new gap function is proposed, and an equivalent optimization problem for the vector variational inequality is also provided. Under some suitable conditions, we prove that the gap function is directionally differentiable and that any point satisfying the first-order necessary optimality condition for the equivalent optimization problem solves the vector variational inequality. As an application, we use the new gap function to reformulate a stochastic vector variational inequality as a deterministic optimization problem. We solve this optimization problem by employing the sample average approximation method. The convergence of optimal solutions of the approximation problems is also investigated.

1. Introduction

The vector variational inequality (VVI for short), which was first proposed by Giannessi [1], has been widely investigated by many authors (see [2–9] and the references therein). VVI can be used to model a range of vector equilibrium problems in economics, traffic networks, and migration equilibrium problems (see [1]).

One approach for solving a VVI is to transform it into an equivalent optimization problem by using a gap function. A gap function was first introduced to study optimization problems and has become a powerful tool in the study of convex optimization problems. Also a gap function was introduced in the study of scalar variational inequalities. It can reformulate a scalar variational inequality as an equivalent optimization problem, and so some effective solution methods and algorithms for optimization problems can be used to find solutions of variational inequalities. Recently, many authors extended the theory of gap functions to VVI and vector equilibrium problems (see [2, 4, 6–9]). In this paper, we present a new gap function for VVI and reformulate it as an equivalent optimization problem. We also prove that the gap function is directionally differentiable and that any point satisfying the first-order necessary optimality condition for the equivalent optimization problem solves the VVI.

In many practical problems, problem data will involve some uncertain factors. In order to reflect the uncertainties, stochastic vector variational inequalities are needed. Recently, stochastic scalar variational inequalities have received a lot of attention in the literature (see [10–20]). The ERM (expected residual minimization) method was proposed by Chen and Fukushima [11] in the study of stochastic complementarity problems. They formulated a stochastic linear complementarity problem (SLCP) as a minimization problem which minimizes the expectation of a NCP function (also called a residual function) of SLCP and regarded a solution of the minimization problem as a solution of SLCP. This method is the so-called expected residual minimization method. Following the ideas of Chen and Fukushima [11], Zhang and Chen [20] considered stochastic nonlinear complementary problems. Luo and Lin [18, 19] generalized the expected residual minimization method to solve a stochastic linear and/or nonlinear variational inequality problem. However, in comparison to stochastic scalar variational inequalities, there are very few results in the literature on stochastic vector variational inequalities. In this paper, we consider a deterministic reformulation for the stochastic vector variational inequality (SVVI). Our focus is on the expected residual minimization (ERM) method for the stochastic vector variational inequality. It is well known that VVI is more complicated than a variational inequality, and they model many practical problems. Therefore, it is meaningful and interesting to study stochastic vector variational inequalities.

The rest of this paper is organized as follows. In Section 2, some preliminaries are given. In Section 3, a new gap function for VVI is constructed and some suitable conditions are given to ensure that the new gap function is directionally differentiable and that any point satisfying the first-order necessary condition of optimality for the new optimization problem solves the vector variational inequality. In Section 4, the stochastic VVI is presented and the new gap function is used to reformulate SVVI as a deterministic optimization problem.

2. Preliminaries

In this section, we will introduce some basic notations and preliminary results.

Throughout this paper, denote by 𝑥^𝑇 the transpose of a vector or matrix 𝑥, by |⋅| the Euclidean norm of a vector or matrix, and by ⟨⋅,⋅⟩ the inner product in ℝ^𝑛. Let 𝐾 be a nonempty, closed, and convex set of ℝ^𝑛,𝐹_𝑖∶ℝ^𝑛→ℝ^𝑛 (𝑖=1,…,𝑝) mappings, and 𝐹∶=(𝐹₁,…,𝐹_𝑝)^𝑇. The vector variational inequality is to find a vector 𝑥^∗∈𝐾 such that 𝐹₁𝑥^∗,𝑦−𝑥^∗,…,𝐹_𝑝𝑥^∗,𝑦−𝑥^∗𝑇∉−intℝ^𝑝₊,∀𝑦∈𝐾, (1) where ℝ^𝑝₊ is the nonnegative orthant of ℝ^𝑝 and intℝ^𝑝₊ denotes the interior of ℝ^𝑝₊. Denote by 𝑆 the solution set of VVI (1) and by 𝑆_𝜉 the solution set of the following scalar variational inequality (VI_𝜉): find a vector 𝑥^∗∈𝐾, such that ^𝑝_𝑗=1𝜉_𝑗𝐹_𝑗𝑥^∗,𝑦−𝑥^∗≥0,∀𝑦∈𝐾, (2) where 𝜉∈𝐵∶={𝜉∈ℝ^𝑝₊∶∑^𝑝_𝑗=1𝜉_𝑗=1}.

Definition 1. A function 𝜑_𝜉∶𝐾→ℝ is said to be a gap function for VI_𝜉 (2) if it satisfies the following properties: (i) 𝜑_𝜉(𝑥)≥0,forall𝑥∈𝐾; (ii) 𝜑_𝜉(𝑥^∗)=0 iff 𝑥^∗ solves VI_𝜉 (2).

Definition 2. A function 𝜓∶𝐾→ℝ is said to be a gap function for VVI (1) if it satisfies the following properties: (i) 𝜓(𝑥)≥0,forall𝑥∈𝐾; (ii) 𝜓(𝑥^∗)=0 iff 𝑥^∗ solves VVI (1).

Suppose that 𝐺 is an 𝑛×𝑛 symmetric positive definite matrix. Let 𝜑_𝜉(𝑥)∶=max_𝑦∈𝐾^𝑝_𝑗=1𝜉_𝑗𝐹_𝑗(𝑥)−1,𝑥−𝑦2𝑥−𝑦²_𝐺, (3) where |𝑥|²_𝐺∶=⟨𝑥,𝐺𝑥⟩. Note that 𝜆_min|𝑥|≤|𝑥|_𝐺≤𝜆_max|𝑥|, (4) where 𝜆_min and 𝜆_max are the smallest and largest eigenvalues of 𝐺, respectively. It was shown in [21] that the maximum in (3) is attained at 𝐻_𝜉(𝑥)∶=Proj_𝐾,𝐺𝑥−𝐺^𝑝−1_𝑗=1𝜉_𝑗𝐹_𝑗(𝑥), (5) where Proj_𝐾,𝐺(𝑥) is the projection of the point 𝑥 onto the closed convex set 𝐾 with respect to the norm |⋅|_𝐺. Thus, 𝜑_𝜉(𝑥)=^𝑝_𝑗=1𝜉_𝑗𝐹_𝑗(𝑥),𝑥−𝐻_𝜉(𝑥−1)2𝑥−𝐻_𝜉(𝑥)²_𝐺. (6)

Lemma 3. The projection operator 𝑃𝑟𝑜𝑗_𝐾,𝐺(⋅) is nonexpansive; that is, 𝑃𝑟𝑜j_𝐾,𝐺(𝑥)−𝑃𝑟𝑜𝑗_𝐾,𝐺(𝑦)_𝐺≤𝑥−𝑦_𝐺,∀𝑥,𝑦∈ℝ^𝑛. (7)

Lemma 4. The function 𝜑_𝜉 is a gap function for 𝑉𝐼_𝜉 (2) and 𝑥^∗∈𝐾 solves VI_𝜉 (2) iff it solves the following optimization problem: min_𝑥∈𝐾𝜑_𝜉(𝑥). (8)

The gap function 𝜑_𝜉 is also called the regularized gap function for VI_𝜉. When 𝐹_𝑗 (𝑗=1,…,𝑝) are continuously differentiable, we have the following results.

Lemma 5. If 𝐹_𝑗 (𝑗=1,…,𝑝) are continuously differentiable, then 𝜑_𝜉 is also continuously differentiable in 𝑥, and its gradient is given by ∇_𝑥𝜑_𝜉(𝑥)=^𝑝_𝑗=1𝜉_𝑗𝐹_𝑗(𝑥)−^𝑝_𝑗=1𝜉_𝑗∇_𝑥𝐹_𝑗(𝑥)−𝐺𝐻_𝜉(𝑥)−𝑥. (9)

Lemma 6. Assume that 𝐹_𝑗 are continuously differentiable and that the Jacobian matrixes ∇_𝑥𝐹_𝑗(𝑥) are positive definite for all 𝑥∈𝐾 (𝑗=1,…,𝑝). If 𝑥 is a stationary point of problem (8), that is, ∇_𝑥𝜑_𝜉(𝑥),𝑦−𝑥≥0,∀𝑦∈𝐾, (10) then it solves 𝑉𝐼_𝜉 (2).

3. A New Gap Function for VVI and Its Properties

In this section, based on the regularized gap function 𝜑_𝜉 for VI_𝜉 (2), we construct a new gap function for VVI (1) and establish some properties under some mild conditions.

Let 𝜓(𝑥)∶=min_𝜉∈𝐵𝜑_𝜉(𝑥). (11) Before showing that 𝜓 is a gap function for VVI, we first present a useful result.

Lemma 7. The following assertion is true: 𝑆=∪_𝜉∈𝐵𝑆_𝜉. (12)

Proof. Suppose that 𝑥^∗∈∪_𝜉∈𝐵𝑆_𝜉. Then, there exists a 𝜉∈𝐵 such that 𝑥^∗∈𝑆_𝜉 and ^𝑝_𝑗=1𝜉_𝑗𝐹_𝑗𝑥^∗,𝑦−𝑥^∗=^𝑝_𝑗=1𝜉_𝑗𝐹_𝑗𝑥^∗,𝑦−𝑥^∗≥0,∀𝑦∈𝐾. (13) For any fixed 𝑦∈𝐾, since 𝜉∈𝐵, there exists a 𝑗∈{1,…,𝑝} such that 𝐹_𝑗𝑥^∗,𝑦−𝑥^∗≥0, (14) and so 𝐹₁𝑥^∗,𝑦−𝑥^∗,…,𝐹_𝑝𝑥^∗,𝑦−𝑥^∗𝑇∉−intℝ^𝑝₊. (15) Thus, we have 𝐹₁𝑥^∗,𝑦−𝑥^∗,…,𝐹_𝑝𝑥^∗,𝑦−𝑥^∗𝑇∉−intℝ^𝑝₊,∀𝑦∈𝐾. (16) This implies that 𝑥^∗∈𝑆 and ∪_𝜉∈𝐵𝑆_𝜉⊂𝑆.
Conversely, suppose that 𝑥^∗∈𝑆. Then, we have 𝐹₁𝑥^∗,𝑦−𝑥^∗,…,𝐹_𝑝𝑥^∗,𝑦−𝑥^∗𝑇∉−intℝ^𝑝₊,∀𝑦∈𝐾, (17) and so 𝐹₁𝑥^∗,𝑦−𝑥^∗,…,𝐹_𝑝𝑥^∗,𝑦−𝑥^∗𝑇∩∶∀𝑦∈𝐾−intℝ^𝑝₊=∅. (18) Since 𝐾 is convex, from Theorems 11.1 and 11.3 of [22], it follows that inf_𝑦∈𝐾^𝑝_𝑗=1𝑏_𝑗𝐹_𝑗𝑥^∗,𝑦−𝑥^∗≥sup_{𝑦∈−intℝ𝑝+}𝑏,𝑦, (19) where 𝑏=(𝑏₁,…,𝑏_𝑝)≠0. Moreover, we have 𝑏>0. In fact, if 𝑏_𝑗<0 for some 𝑗∈{1,…,𝑝}, then we have sup_{𝑦∈(−intℝ𝑝+)}⟨𝑏,𝑦⟩=+∞. (20) On the other hand, inf_𝑦∈𝐾^𝑝_𝑗=1𝑏_𝑗𝐹_𝑗𝑥^∗,𝑦−𝑥^∗≤^𝑝_𝑗=1𝑏_𝑗𝐹_𝑗𝑥^∗,𝑥^∗−𝑥^∗=0<sup_{𝑦∈(−intℝ𝑝+)}⟨𝑏,𝑦⟩, (21) which is a contradiction. Thus, 𝑏>0. This implies that, for any 𝑧∈𝐾, ^𝑝_𝑗=1𝑏_𝑗𝐹_𝑗𝑥^∗,𝑧−𝑥^∗≥inf_𝑦∈𝐾^𝑝_𝑗=1𝑏_𝑗𝐹_𝑗𝑥^∗,𝑦−𝑥^∗≥sup_{𝑦∈−intℝ𝑝+}𝑏,𝑦=0. (22) Taking 𝜉=𝑏/(∑^𝑝_𝑗=1𝑏_𝑗), then 𝜉∈𝐵 and ^𝑝_𝑗=1𝜉_𝑗𝐹_𝑗𝑥^∗,𝑧−𝑥^∗≥0,∀𝑧∈𝐾, (23) which implies that 𝑥^∗∈𝑆_𝜉 and 𝑆⊂∪_𝜉∈𝐵𝑆_𝜉.
This completes the proof.

Now, we can prove that 𝜓 is a gap function for VVI (1).

Theorem 8. The function 𝜓 given by (11) is a gap function for VVI (1). Hence, 𝑥^∗∈𝐾 solves VVI (1) iff it solves the following optimization problem: min_𝑥∈𝐾𝜓(𝑥). (24)

Proof. Note that for any 𝜉∈𝐵, 𝜑_𝜉(𝑥) given by (3) is a gap function for VI_𝜉 (2). It follows from Definition 1 that 𝜑_𝜉(𝑥)≥0 for all 𝑥∈𝐾 and hence 𝜓(𝑥)=min_𝜉∈𝐵𝜑_𝜉(𝑥)≥0 for all 𝑥∈𝐾.
Assume that 𝜓(𝑥^∗)=0 for some 𝑥^∗∈𝐾. From (6), it is easy to see that 𝜑_𝜉(𝑥^∗) is continuous in 𝜉. Since 𝐵 is a closed and bounded set, there exists a vector 𝜉∈𝐵 such that 𝜓(𝑥^∗)=𝜑_𝜉(𝑥^∗), which implies that 𝜑_𝜉(𝑥^∗)=0 and 𝑥^∗∈𝑆_𝜉. It follows from Lemma 7 that 𝑥^∗ solves VVI (1).
Suppose that 𝑥^∗ solves VVI (1). From Lemma 7, it follows that there exists a vector 𝜉∈𝐵 such that 𝑥^∗∈𝑆_𝜉 and so 𝜑_𝜉(𝑥^∗)=0. Since, for all 𝜉∈𝐵, 𝜑_𝜉(𝑥^∗)≥0, we have 𝜓(𝑥^∗)=min_𝜉∈𝐵𝜑_𝜉(𝑥^∗)=0.
Thus, (11) is a gap function for VVI (1). The last assertion is obvious from the definition of gap function.
This completes the proof.

Since 𝜓 is constructed based on the regularized gap function for VI_𝜉, we wish to call it a regularized gap function for VVI (1). Theorem 8 indicates that in order to get a solution of VVI (1), we only need to solve problem (24). In what follows, we will discuss some properties of the regularized gap function 𝜓.

Theorem 9. If 𝐹_𝑗 (𝑗=1,…,𝑝) are continuously differentiable, then the regularized gap function 𝜓 is directionally differentiable in any direction 𝑑∈ℝ^𝑛, and its directional derivative 𝜓^′(𝑥;𝑑) is given by 𝜓^′(𝑥;𝑑)=inf_{(𝑥)𝜉∈𝐵}∇_𝑥𝜑_𝜉(𝑥),𝑑, (25) where 𝐵(𝑥)∶={𝜉∈𝐵∶𝜑_𝜉(𝑥)=min_𝜉∈𝐵𝜑_𝜉(𝑥)}.

Proof. It follows since the projection operator is nonexpansive that 𝜑_𝜉(𝑥) is continuous in (𝜉,𝑥). Thus, 𝐵(𝑥) is nonempty for any 𝑥∈𝐾. From Lemma 5, it follows that 𝜑_𝜉(𝑥) is continuously differentiable in 𝑥 and that ∇_𝑥𝜑_𝜉(𝑥)=^𝑝_𝑗=1𝜉_𝑗𝐹_𝑗(𝑥)−^𝑝_𝑗=1𝜉_𝑗∇_𝑥𝐹_𝑗(𝑥)−𝐺𝐻_𝜉(𝑥)−𝑥 (26) is continuous in (𝜉,𝑥). It follows from Theorem 1 of [23] that 𝜓 is directionally differentiable in any direction 𝑑∈ℝ^𝑛, and its directional derivative 𝜓^′(𝑥;𝑑) is given by 𝜓^′(𝑥;𝑑)=inf_{(𝑥)𝜉∈𝐵}∇_𝑥𝜑_𝜉(𝑥),𝑑. (27) This completes the proof.

By the directional differentiability of 𝜓 shown in Theorem 9, the first-order necessary condition of optimality for problem (24) can be stated as 𝜓^′𝑥;𝑦−𝑥≥0,∀𝑦∈𝐾. (28) If one wishes to solve VVI (1) via the optimization problem (24), we need to obtain its global optimal solution. However, since the function 𝜓 is in general nonconvex, it is difficult to find a global optimal solution. Fortunately, we can prove that any point satisfying the first-order condition of optimality becomes a solution of VVI (1). Thus, existing optimization algorithms can be used to find a solution of VVI (1).

Theorem 10. Assume that 𝐹_𝑗 are continuously differentiable and that the Jacobian matrixes ∇_𝑥𝐹_𝑗(𝑥) are positive definite for all 𝑥∈𝐾 (𝑗=1,…,𝑝). If 𝑥^∗∈𝐾 satisfies the first-order condition of optimality (28), then 𝑥^∗ solves VVI (1).

Proof. It follows from Theorem 9 and (28) that, for some 𝜉∈𝐵(𝑥^∗), ∇^∗_𝑥𝜑_𝜉𝑥^∗,𝑦−𝑥^∗≥0 (29) holds for any 𝑦∈𝐾. This implies that 𝑥^∗ is a stationary point of problem (8). It follows from Lemma 6 that 𝑥^∗ solves VI_𝜉. From Lemma 7, we see that 𝑥^∗ solves VVI. This completes the proof.

From Theorems 8 and 10, it is easy to get the following corollary.

Corollary 11. Assume that the conditions in Theorem 10 are all satisfied. If 𝑥^∗∈𝐾 satisfies the first-order condition of optimality (28), then 𝑥^∗ is a global optimal solution of problem (24).

4. Stochastic Vector Variational Inequality

In this section, we consider the stochastic vector variational inequality (SVVI). First, we present a deterministic reformulation for SVVI by employing the ERM method and the regularized gap function. Second, we solve this reformulation by the SAA method.

In most important practical applications, the functions 𝐹_𝑗 (𝑗=1,…,𝑝) always involve some random factors or uncertainties. Let (Ω,ℱ,𝑃) be a probability space. Taking the randomness into account, we get a stochastic vector variational inequality problem (SVVI): find a vector 𝑥^∗∈𝐾 such that 𝐹₁𝑥^∗,𝜔,𝑦−𝑥^∗,…,𝐹_𝑝𝑥^∗,𝜔,𝑦−𝑥^∗𝑇∉−intℝ^𝑝₊,∀𝑦∈𝐾,a.s., (30) where 𝐹_𝑗∶ℝ^𝑛×Ω→ℝ^𝑛 are mappings and a.s. is the abbreviation for “almost surely" under the given probability measure 𝑃.

Because of the random element 𝜔, we cannot generally find a vector 𝑥^∗∈𝐾 such that (30) holds almost surely. That is, (30) is not well defined if we think of solving (30) before knowing the realization 𝜔. Therefore, in order to get a reasonable resolution, an appropriate deterministic reformulation for SVVI becomes an important issue in the study of the considered problem. In this section, we will employ the ERM method to solve (30).

Define 𝜓(𝑥,𝜔)∶=min_𝜉∈𝐵𝜑_𝜉(𝑥,𝜔), (31) where 𝜑_𝜉(𝑥,𝜔)∶=max_𝑦∈𝐾^𝑝_𝑗=1𝜉_𝑗𝐹_𝑗()−1𝑥,𝜔,𝑥−𝑦2𝑥−𝑦²_𝐺. (32) The maximum in (32) is attained at 𝐻_𝜉(𝑥,𝜔)∶=Proj_𝐾,𝐺𝑥−𝐺^𝑝−1_𝑗=1𝜉_𝑗𝐹_𝑗(),𝜑𝑥,𝜔_𝜉(𝑥,𝜔)=^𝑝_𝑗=1𝜉_𝑗𝐹_𝑗(𝑥,𝜔),𝑥−𝐻_𝜉(−1𝑥,𝜔)2𝑥−𝐻_𝜉(𝑥,𝜔)²_𝐺. (33)(34) The ERM reformulation is given as follows: min_𝑥∈𝐾Θ(𝑥)∶=𝔼𝜓(𝑥,𝜔), (35) where 𝔼 is the expectation operator.

Note that the objective function Θ(𝑥) contains the mathematical expectation. In practical applications, it is in general very difficult to calculate 𝔼[𝜓(𝑥,𝜔)] in a closed form. Thus, we will have to approximate it through discretization. One of the most popular discretization approaches is the sample average approximation method. In general, for an integrable function 𝜙∶Ω→ℝ, we approximate the expected value 𝔼[𝜙(𝜔)] with the sample average (1/𝑁_𝑘)∑_{𝜔𝑖∈Ω𝑘}𝜙(𝜔_𝑖), where 𝜔₁,…,𝜔_𝑁𝑘 are independently and identically distributed random samples of 𝜔 and Ω_𝑘∶={𝜔₁,…,𝜔_𝑁𝑘}. By the strong law of large numbers, we get the following lemma.

Lemma 12. If 𝜙(𝜔) is integrable, then lim_𝑘→∞1𝑁_{𝑘𝜔𝑖∈Ω𝑘}𝜙𝜔_𝑖=𝔼𝜙(𝜔) (36) holds with probability one.

Let Θ_𝑘(𝑥)∶=(1/𝑁_𝑘)∑_{𝜔𝑖∈Ω𝑘}𝜓(𝑥,𝜔_𝑖). Applying the above technique, we get the following approximation of (35): min_𝑥∈𝐾Θ_𝑘(𝑥). (37) In the rest of this section, we focus on the case 𝐹_𝑗(𝑥,𝜔)∶=𝑀_𝑗(𝜔)𝑥+𝑄_𝑗(𝜔) (38) (𝑗=1,…,𝑝), where 𝑀_𝑗∶Ω→ℝ^𝑛×𝑛 and 𝑄_𝑗∶Ω→ℝ^𝑛 are measurable functions such that 𝔼𝑀_𝑗(𝜔)²<+∞,𝔼𝑄(𝜔)_𝑗²<+∞,𝑗=1,…,𝑝. (39) This condition implies that 𝔼^𝑝_𝑗=1𝑀_𝑗(𝜔)²𝔼<+∞,_𝑝𝑗=1𝑀_𝑗(𝜔)^𝑝_𝑗=1𝑄_𝑗(𝜔)<+∞, (40) and, for any scalar 𝑐, 𝔼^𝑝_𝑗=1𝑀_𝑗(𝜔)𝑐+𝑄_𝑗(𝜔)<+∞. (41)

The following results will be useful in the proof of the convergence result.

Lemma 13. Let 𝑓,𝑔∶ℝ^𝑝→ℝ₊ be continuous. If min_𝜉∈𝐵𝑓(𝜉)<+∞ and min_𝜉∈𝐵𝑔(𝜉)<+∞, then min_𝜉∈𝐵𝑓(𝜉)−min_𝜉∈𝐵𝑔(𝜉)≤max_𝜉∈𝐵𝑓(𝜉)−𝑔(𝜉). (42)

Proof. Without loss of generality, we assume that min_𝜉∈𝐵𝑓(𝜉)≤min_𝜉∈𝐵𝑔(𝜉). Let 𝜉 minimize 𝑓 and 𝜉̂ minimize 𝑔, respectively. Hence, 𝑓(𝜉)≤𝑔(𝜉̂) and 𝑔(𝜉̂)≤𝑔(𝜉). Thus min_𝜉∈𝐵𝑓(𝜉)−min_𝜉∈𝐵𝑔(𝜉)=𝑔𝜉̂−𝑓𝜉≤𝑔𝜉−𝑓𝜉≤max_𝜉∈𝐵𝑓(𝜉)−𝑔(𝜉). (43)
This completes the proof.

Lemma 14. When 𝐹_𝑗(𝑥,𝜔)=𝑀_𝑗(𝜔)𝑥+𝑄_𝑗(𝜔) (𝑗=1,…,𝑝), the function 𝜑_𝜉(𝑥,𝜔) is continuously differentiable in 𝑥 almost surely, and its gradient is given by ∇_𝑥𝜑_𝜉(𝑥,𝜔)=^𝑝_𝑗=1𝜉_𝑗𝐹_𝑗()−𝑥,𝜔_𝑝𝑗=1𝜉_𝑗𝑀_𝑗(𝜔)−𝐺𝐻_𝜉(𝑥,𝜔)−𝑥. (44)

Proof. The proof is the same as that of Theorem  3.2 in [21], so we omit it here.

Lemma 15. For any 𝑥∈𝐾, one has 𝑥−𝐻_𝜉(≤2𝑥,𝜔)𝜆^𝑝_min𝑗=1𝐹_𝑗(𝑥,𝜔). (45)

Proof. For any fixed 𝜔∈Ω,  𝜑_𝜉(𝑥,𝜔) is the gap function of the following scalar variational inequality: find a vector 𝑥^∗∈𝐾, such that ^𝑝_𝑗=1𝜉_𝑗𝐹_𝑗𝑥^∗,𝜔,𝑦−𝑥^∗≥0,∀𝑦∈𝐾. (46) Hence, 𝜑_𝜉(𝑥,𝜔)≥0 for all 𝑥∈𝐾. From (4) and (34), we have 12𝑥−𝐻_𝜉(𝑥,𝜔)²_𝐺≤^𝑝_𝑗=1𝜉_𝑗𝐹_𝑗(𝑥,𝜔),𝑥−𝐻_𝜉(≤𝑥,𝜔)_𝑝𝑗=1𝜉_𝑗𝐹_𝑗(𝑥,𝜔)𝑥−𝐻_𝜉(≤1𝑥,𝜔)𝜆^𝑝_min𝑗=1𝐹_𝑗(𝑥,𝜔)𝑥−𝐻_𝜉(𝑥,𝜔)_𝐺, (47) and so 𝑥−𝐻_𝜉(𝑥,𝜔)_𝐺≤2𝜆^𝑝_min𝑗=1𝐹_𝑗(𝑥,𝜔). (48) It follows from (4) that 𝑥−𝐻_𝜉(≤1𝑥,𝜔)𝜆_min𝑥−𝐻_𝜉(𝑥,𝜔)_𝐺≤2𝜆^𝑝_min𝑗=1𝐹_𝑗(𝑥,𝜔). (49)
This completes the proof.

Now, we obtain the convergence of optimal solutions of problem (37) in the following theorem.

Theorem 16. Let {𝑥^𝑘} be a sequence of optimal solutions of problem (37). Then, any accumulation point of {𝑥^𝑘} is an optimal solution of problem (35).

Proof. Let 𝑥^∗ be an accumulation point of {𝑥^𝑘}. Without loss of generality, we assume that 𝑥^𝑘 itself converges to 𝑥^∗ as 𝑘 tends to infinity. It is obvious that 𝑥^∗∈𝐾. At first, we will show that lim_𝑘→∞1𝑁_{𝑘𝜔𝑖∈Ω𝑘}𝜓𝑥^𝑘,𝜔_𝑖=𝔼𝜓𝑥^∗,𝜔. (50) From Lemma 12, it suffices to show that lim_𝑘→∞1𝑁_{𝑘𝜔𝑖∈Ω𝑘}𝜓𝑥^𝑘,𝜔_𝑖−1𝑁_{𝑘𝜔𝑖∈Ω𝑘}𝜓𝑥^∗,𝜔_𝑖=0. (51) It follows from Lemma 13 and the mean-value theorem that 1𝑁_{𝑘𝜔𝑖∈Ω𝑘}𝜓𝑥^𝑘,𝜔_𝑖−1𝑁_{𝑘𝜔𝑖∈Ω𝑘}𝜓𝑥^∗,𝜔_𝑖≤1𝑁_{𝑘𝜔𝑖∈Ω𝑘}𝜓𝑥^𝑘,𝜔_𝑖−𝜓𝑥^∗,𝜔_𝑖=1𝑁_{𝑘𝜔𝑖∈Ω𝑘}min_𝜉∈𝐵𝜑_𝜉𝑥^𝑘,𝜔_𝑖−min_𝜉∈𝐵𝜑_𝜉𝑥^∗,𝜔_𝑖≤1𝑁_{𝑘𝜔𝑖∈Ω𝑘}max_𝜉∈𝐵𝜑_𝜉𝑥^𝑘,𝜔_𝑖−𝜑_𝜉𝑥^∗,𝜔_𝑖≤1𝑁_{𝑘𝜔𝑖∈Ω𝑘}max_𝜉∈𝐵∇_𝑥𝜑_𝜉𝑦^𝑘,𝑖,𝜔_𝑖𝑥^𝑘−𝑥^∗, (52) where 𝑦^𝑘,𝑖=𝜆_𝑘,𝑖𝑥^𝑘+(1−𝜆_𝑘,𝑖)𝑥^∗ with 𝜆_𝑘,𝑖∈[0,1]. Because lim_𝑘→+∞𝑥^𝑘=𝑥^∗, there exists a constant 𝐶 such that |𝑥^𝑘|≤𝐶 for each 𝑘. By the definition of 𝑦^𝑘,𝑖, we have |𝑦^𝑘,𝑖|≤𝐶. Hence, for any 𝜉∈𝐵 and 𝜔_𝑖∈Ω_𝑘, ∇_𝑥𝜑_𝜉𝑦^𝑘,𝑖,𝜔_𝑖=^𝑝_𝑗=1𝜉_𝑗𝐹_𝑗𝑦^𝑘,𝑖,𝜔_𝑖−^𝑝_𝑗=1𝜉_𝑗∇_𝑥𝐹_𝑗𝑦^𝑘,𝑖,𝜔_𝑖×𝐻−𝐺_𝜉𝑦^𝑘,𝑖,𝜔_𝑖−𝑦^𝑘,𝑖≤^𝑝_𝑗=1𝜉_𝑗𝐹_𝑗𝑦^𝑘,𝑖,𝜔_𝑖+^𝑝_𝑗=1𝜉_𝑗∇_𝑥𝐹_𝑗𝑦^𝑘,𝑖,𝜔_𝑖+|𝐺×𝐻_𝜉|𝑦^𝑘,𝑖,𝜔_𝑖−𝑦^𝑘,𝑖≤^𝑝_𝑗=1𝐹_𝑗𝑦^𝑘,𝑖,𝜔_𝑖+2𝜆_min^𝑝_𝑗=1∇_𝑥𝐹_𝑗𝑦^𝑘,𝑖,𝜔_𝑖+|𝐺|×^𝑝_𝑗=1𝐹_𝑗𝑦^𝑘,𝑖,𝜔_𝑖≤2𝜆^𝑝_min𝑗=1𝑀_𝑗𝜔_𝑖+2𝜆_min|𝐺|×+1_𝑝𝑗=1𝑀_𝑗𝜔_𝑖𝐶+𝑄_𝑗𝜔_𝑖=2𝐶𝜆_min^𝑝_𝑗=1𝑀_𝑗𝜔_𝑖²+^𝑝_𝑗=1𝑀_𝑗𝜔_𝑖𝐶+𝑄_𝑗𝜔_𝑖2𝜆_min|𝐺|+2+1𝜆_min^𝑝_𝑗=1𝑀_𝑗𝜔_𝑖^𝑝_𝑗=1𝑄_𝑗𝜔_𝑖, (53) where the second inequality is from Lemma 15. Thus, 1𝑁_{𝑘𝜔𝑖∈Ω𝑘}𝜓𝑥^𝑘,𝜔_𝑖−1𝑁_{𝑘𝜔𝑖∈Ω𝑘}𝜓𝑥^∗,𝜔_𝑖≤1𝑁_{𝑘𝜔𝑖∈Ω𝑘}max_𝜉∈𝐵∇_𝑥𝜑_𝜉𝑦^𝑘,𝑖,𝜔_𝑖𝑥^𝑘−𝑥^∗≤2𝐶𝜆_min𝑥^𝑘−𝑥^∗1𝑁_{𝑘𝜔𝑖∈Ω𝑘}^𝑝_𝑗=1𝑀_𝑗𝜔_𝑖²+𝑥^𝑘−𝑥^∗1𝑁_{𝑘𝜔𝑖∈Ω𝑘}^𝑝_𝑗=1𝑀_𝑗𝜔_𝑖𝐶+𝑄_𝑗𝜔_𝑖×2𝜆_min|𝐺|+2+1𝜆_min𝑥^𝑘−𝑥^∗1𝑁_{𝑘𝜔𝑖∈Ω𝑘}^𝑝_𝑗=1𝑀_𝑗𝜔_𝑖×^𝑝_𝑗=1𝑄_𝑗𝜔_𝑖. (54) From (40) and (41), each term in the last inequality above converges to zero, and so lim_𝑘→∞1𝑁_{𝑘𝜔𝑖∈Ω𝑘}𝜓𝑥^𝑘,𝜔_𝑖−1𝑁_{𝑘𝜔𝑖∈Ω𝑘}𝜓𝑥^∗,𝜔_𝑖=0. (55) Since lim_𝑘→∞1𝑁_{𝑘𝜔𝑖∈Ω𝑘}𝜓𝑥^∗,𝜔_𝑖=𝔼𝜓𝑥^∗,𝜔, (56) (50) is true.
Now, we are in the position to show that 𝑥^∗ is a solution of problem (35).
Since 𝑥^𝑘 solves problem (37) for each 𝑘, we have that, for any 𝑥∈𝐾, 1𝑁_{𝑘𝜔𝑖∈Ω𝑘}𝜓𝑥^𝑘,𝜔_𝑖≤1𝑁_{𝑘𝜔𝑖∈Ω𝑘}𝜓𝑥,𝜔_𝑖. (57) Letting 𝑘→∞ above, we get from Lemma 12 and (50) that 𝔼𝜓𝑥^∗,𝜔≤𝔼𝜓(𝑥,𝜔), (58) which means that 𝑥^∗ is an optimal solution of problem (35). This completes the proof.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11171237, 11101069, and 71101099), by the Construction Foundation of Southwest University for Nationalities for the subject of applied economics (2011XWD-S0202), and by Sichuan University (SKQY201330).